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A RCHITECTURAL S TRUCTURES : Mechanics of Materials F ORM, B EHAVIOR, AND D ESIGN ARCH 331 • MECHANICS • MATERIALS D R. A NNE N ICHOLS S PRING 2019 lecture six mechanics www.carttalk.com of materials Mechanics of Materials 1 Architectural Structures F2009abn Lecture 6 ARCH 331 Mechanics of Materials 2 Foundations Structures F2008abn Lecture 5 ARCH 331 Mechanics of Materials Knowledge Required • external loads and their • material properties effect on deformable bodies • member cross sections • use it to answer question if structure • ability of a material to resist breaking meets requirements of • structural elements that resist excessive – stability and equilibrium – deflection – strength and stiffness – deformation • other principle building requirements • economy, functionality and aesthetics Mechanics of Materials 3 Foundations Structures F2008abn Mechanics of Materials 4 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 1

Stress Problem Solving • stress is a term for the intensity of a 1. STATICS: force, like a pressure equilibrium of external forces, • internal or applied internal forces, stresses 2. GEOMETRY: • force per unit area cross section properties, deformations and conditions of geometric fit, strains P   stress f 3. MATERIAL PROPERTIES: A stress-strain relationship for each material obtained from testing Mechanics of Materials 6 Foundations Structures F2008abn Mechanics of Materials 5 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Design Design (cont) • • we ’ d like materials have a critical stress value  f F where they could break or yield actual allowable – ultimate stress • stress distribution may – yield stress vary: average acceptance – compressive stress vs. failure • uniform distribution – fatigue strength exists IF the member is – (creep & temperature) loaded axially (concentric) Mechanics of Materials 7 Foundations Structures F2008abn Mechanics of Materials 8 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 2

Scale Effect Normal Stress (direct) • model scale • normal stress is normal to the cross section – material weights by volume, small section areas – stressed area is • structural scale perpendicular to the load – much more material weight, bigger section areas P  • scale for strength is not f t or c   A  proportional:  3 L   L 2 L Mechanics of Materials 10 Foundations Structures F2008abn Mechanics of Materials 9 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Shear Stress Bearing Stress • • stress parallel to a surface stress on a surface by contact in compression P P P P     f v f p   A td    A td  ave Mechanics of Materials 11 Foundations Structures F2008abn Mechanics of Materials 12 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 3

Bending Stress Torsional Stress • • normal stress caused by bending shear stress caused by twisting  Mc M T    f b f v     I S J   Mechanics of Materials 13 Foundations Structures F2008abn Mechanics of Materials 14 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Structures and Shear Bolts • • what structural elements see shear? connected members in tension cause – shear stress beams – bolts connections – splices – slabs – footings • connected members in V – walls  compression cause • wind bearing stress • seismic loads Mechanics of Materials 15 Foundations Structures F2008abn Mechanics of Materials 16 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 4

Single Shear Double Shear • • seen when 2 members are connected seen when 3 members are connected • two areas F= P P P 2 2    f P P v    2 2 A A d f 4  v 2 A d 4 Mechanics of Materials 17 Foundations Structures F2008abn Mechanics of Materials 18 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Strain Bolt Bearing Stress • materials deform • compression & contact • axially loaded materials change • projected area F= length • bending materials deflect  • STRAIN: P P L      f strain – change in length p A td L projected over length + UNITLESS Mechanics of Materials 19 Foundations Structures F2008abn Mechanics of Materials 20 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 5

Shearing Strain Shearing Strain • deformations • deformations  s  with shear with torsion L • parallelogram • twist • change in angles • change in angle of line    • stress: • stress:            • strain: • strain: s L tan L – unitless (radians) – unitless (radians) Mechanics of Materials 21 Foundations Structures F2008abn Mechanics of Materials 22 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Load and Deformation Material Behavior • for stress, need P & A • every material has its own response • for strain, need  & L – 10,000 psi – L = 10 in – how? – Douglas Fir vs. – TEST with load and steel? measure – plot P/A vs.  Mechanics of Materials 23 Foundations Structures F2008abn Mechanics of Materials 24 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 6

Behavior Types Behavior Types • ductile - “ necking ” • brittle • true stress P f  A • engineering stress • semi-brittle – (simplified) P f  A o Mechanics of Materials 25 Foundations Structures F2008abn Mechanics of Materials 26 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Hooke ’ s Law Stress to Strain • important to us in -  diagrams: • straight line has constant slope f – straight section • Hooke ’ s Law f – LINEAR-ELASTIC E  E   f – recovers shape 1 (no permanent  • E deformation) – Modulus of elasticity – Young ’ s modulus – units just like stress Mechanics of Materials 27 Foundations Structures F2008abn Mechanics of Materials 28 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 7

Isotropy & Anisotropy Stiffness • ISOTROPIC • ability to resist strain – materials with E same at f u any direction of loading • steels – ex. steel – same E – different • ANISOTROPIC yield points – materials with different E – different at any direction of loading ultimate strength – ex. wood is orthotropic Mechanics of Materials 30 Foundations Structures F2008abn Mechanics of Materials 29 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Elastic, Plastic, Fatigue Plastic Behavior • elastic springs back • ductile • plastic has permanent deformation at yield stress • fatigue caused by reversed loading cycles Mechanics of Materials 31 Foundations Structures F2008abn Mechanics of Materials 32 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 8

Poisson ’ s Ratio Lateral Strain • or “ what happens to the cross section • constant relationship between with axial stress ” longitudinal strain and lateral strain  f x   x  lateral strain        y z E   axial strain x x    f x 0 f f      y z • strain in lateral direction y z E – negative    y z – equal for isometric materials    • sign! 0 0 . 5 Mechanics of Materials 33 Foundations Structures F2008abn Mechanics of Materials 34 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Calculating Strain Orthotropic Materials • from Hooke ’ s law • non-isometric  E   • directional values of f E and  • substitute  • ex: P   E – plywood A L – laminates PL – polymer   • get  composites AE Mechanics of Materials 35 Foundations Structures F2008abn Mechanics of Materials 36 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 9

Maximum Stresses Stress Concentrations • why we use f ave • if we need to know • increase in stress at where max f and f v F changes in geometry happen: – sharp notches P  – holes       f 0 cos 1 max – corners A o         45 cos sin 0 . 5 P f   max f  – v max 2 A 2 o Mechanics of Materials 38 Foundations Structures F2008abn Mechanics of Materials 37 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Maximum Stresses Deformation Relationships • physical movement – axially (same or zero) – rotations from axial changes          aluminum steel 20 kN PL   relates  to P • AE Mechanics of Materials 40 Foundations Structures F2008abn Mechanics of Materials 39 Foundations Structures F2008abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 10